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Cubic Semisymmetric Graph

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A cubic semisymmetric graph is a graph that is both cubic (i.e., regular of vertex degree 3) and semisymmetric (i.e., edge-transitive but not vertex-transitive).

CubicSemisymmetricGraphs

The four smallest cubic semisymmetric graphs are illustrated above. The smallest of these is Gray graph on 54 vertices, the next smallest is the Iofinova-Ivanov graph on 110 vertices (Iofinova and Ivanov 2002, Marušič et al. 2005), the third is the Ljubljana graph (Conder et al. 2002), and the fourth is the Tutte 12-cage.

Some cubic semisymmetric graphs are summarized in the following table.

|V|graph
54Gray graph
110110-Iofinova-Ivanov graph
112Ljubljana graph
126Tutte 12-cage
182182-Iofinova-Ivanov graph
506506-Iofinova-Ivanov graph
990990-Iofinova-Ivanov graph

See also

Cubic Graph, Iofinova-Ivanov Graphs, Ljubljana Graph, Semisymmetric Graph, Tutte 12-Cage

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References

Bouwer, I. A. "On Edge but Not Vertex Transitive Regular Graphs." J. Combin. Th. Ser. B 12, 32-40, 1972.Conder, M.; Malnič, A.; Marušič, D.; Pisanski, T.; and Potočnik, P. "The Edge-Transitive but Not Vertex-Transitive Cubic Graph on 112 Vertices." J. Graph Th. 50, 25-42, 2005. https://doi.org/10.1002/jgt.20089.Iofinova, M. E. and Ivanov, A. A. "Bi-Primitive Cubic Graphs." In Investigations in Algebraic Theory of Combinatorial Objects. (Ed. I. A. Faradžev, A. A. Ivanov, M. H. Klin, and A. J. Woldar). Dordrecht, Netherlands: Kluwer, pp. 459-472, 1994.Ivanov, A. V. "On Edge but Not Vertex Transitive Regular Graphs." In Combinatorial Design Theory (Ed. C. J. Colbourn and R. Mathon). Amsterdam, Netherlands: North-Holland, pp. 273-285, 1987.Marušič, D.; Pisanski, T.; and Wilson, S. "The Genus of the Gray Graph Is 7." Europ. J. Combin. 26, 377-385, 2005.

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Cubic Semisymmetric Graph

Cite this as:

Weisstein, Eric W. "Cubic Semisymmetric Graph." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CubicSemisymmetricGraph.html

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